Yes, you can use either the FileBrowser app to access the hard drive via SMB sharing protocol or the FilesConnect app to access it via the Apple AFP protocol. Both apps will give you Finder-style access to the drive from your iPad.

It's great to use a USB hard drive on a router like the Airport for this purpose, you can avoid leaving a computer on all the time to serve files to the iPad, and you don't need iTunes to transfer media wirelessly.

FileBrowser is cheaper than FilesConnect and has a better interface, but FilesConnect is more powerful and can access Dropbox, Facebook, etc. as well.

You won't need an internet connection, but you will need the local IP address of the router for FileBrowser, which you can get from a router utility like Airport Utility for Mac (if you have an Airport router, for example).

FilesConnect is so good at detecting wireless shares that you won't even need to give it this address.